# Generalized ulam-hyers stability of *C**-Ternary algebra *n*-Homomorphisms for a functional equation

- Won-Gil Park
^{1}and - Jae-Hyeong Bae
^{2}Email author

**2011**:30

https://doi.org/10.1186/1029-242X-2011-30

© Park and Bae; licensee Springer. 2011

**Received: **15 December 2010

**Accepted: **9 August 2011

**Published: **9 August 2011

## Abstract

## Keywords

*n*-homomorphisms

*C**-ternary algebra

## 1. Introduction and preliminaries

*n*-array structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [3]):

- (1)

- (2)
The quark model inspired a particular brand of ternary algebraic systems. The so-called

*Nambu mechnics*is based on such structures [5].

There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and Yang-Baxter equation [4, 6].

A *C**-ternary algebra is a complex Banach space *A*, equipped with a ternary product (*x*, *y*, *z*) α [*x*, *y*, *z*] of *A*^{3} into *A*, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [*x*, *y*, [*z*, *w*, *v*]] = [*x*, [*w*, *z*, *y*], *v*] = [[*x*, *y*, *z*], *w*, *v*], and satisfies ||[*x*, *y*, *z*]|| ≤ ||*x*|| · ||*y*|| · ||*z*|| and ||[*x*, *x*, *x*]|| = ||*x*||^{3} (see [7, 8]). Every left Hilbert *C**-module is a *C**-ternary algebra via the ternary product [*x*, *y*, *z*] := 〈*x*, *y*〉 *z*.

If a *C**-ternary algebra (*A*,[·, ·, ·]) has an identity, i.e., an element *e* ∈ *A* such that *x* = [*x*, *e*, *e*] = [*e*, *e*, *x*] for all *x* ∈ *A*, then it is customary to verify that *A*, endowed with *x* ∘ *y* := [*x*, *e*, *y*] and *x** := [*e*, *x*, *e*], is a unital *C**-algebra. Conversely, if (*A*, ∘) is a unital *C**-algebra, then [*x*, *y*, *z*] := *x* ∘ *y** ∘ *z* makes *A* into a *C**-ternary algebra.

*A*and

*B*be

*C**-ternary algebras. A ℂ-linear mapping

*H*:

*A*→

*B*is called a

*C**-

*ternary algebra homomorphism*if

for all *x*, *y*, *z* ∈ *A*.

**Definition**. Let

*A*and

*B*be

*C**-ternary algebras. A multilinear mapping

*H*:

*A*

^{ n }→

*B*over ℂ is called a

*C**-

*ternary algebra n*-

*homomorphism*if it satisfies

for all *x*_{1}, *y*_{1}, *z*_{1}, · · ·, *x*_{
n
} , *y*_{
n
} , *z*_{
n
} ∈ *A*.

In 1940, Ulam [9] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:

*We are given a group G and a metric group G' with metric ρ*(·, ·). *Given ε* > 0, *does there exist a δ* > 0 *such that if f* : *G* → *G' satisffies ρ*(*f*(*xy*), *f*(*x*) *f*(*y*)) < *δ for all x*, *y* ∈ *G*, *then a homomorphism h* : *G* → *G' exists with ρ*(*f*(*x*), *h*(*x*)) < *ε for all x* ∈ *G ?*

In 1941, Hyers [10] gave the first partial solution to Ulam's question for the case of approximate additive mappings under the assumption that *G*_{1} and *G*_{2} are Banach spaces. Then, Aoki [11] and Bourgin [12] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [13] generalized the theorem of Hyers [10] by considering the stability problem with unbounded Cauchy differences. In 1991, Gajda [14], following the same approach as that by Rassias [13], gave an affirmative solution to this question for *p* > 1. It was shown by Gajda [14] as well as by Rassias and Šemrl [15], that one cannot prove a Rassias-type theorem when *p* = 1. Găvruta [16] obtained the generalized result of Rassias's theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings, for example, Cauchy-Jensen mappings, *k*-additive mappings, invariant means, multiplicative mappings, bounded *n* th differences, convex functions, generalized orthogonality mappings, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations. The instability of characteristic flows of solutions of partial differential equations is related to the Ulam's stability of functional equations [17]-[24]. On the other hand, the authors [25], Park [20] and Kim [26] have contributed studies in respect of the stability problem of ternary homomorphisms and ternary derivations.

## 2. Solution and stability

*X*and

*Y*be real or complex vector spaces and

*n*≥ 2 an integer. For a mapping

*f*:

*X*

^{ n }→

*Y*, consider the functional equation:

We solve the general problem in vector spaces for the *n*-additive mappings satisfying (2.1).

**Theorem 2.1**. *A mapping f* : *X*^{
n
} → *Y satisffies the equation* (2.1) *if and only if the mapping f is n-additive*.

*Proof*. Assume that

*f*satisfies (2.1). Letting

*x*

_{1}=

*y*

_{1}= · · · =

*x*

_{ n }=

*y*

_{ n }= 0 in (2.2), we get

*f*(0, · · ·, 0) = 0. Letting

*y*

_{1}=

*x*

_{2}=

*y*

_{2}= · · · =

*x*

_{ n }=

*y*

_{ n }= 0 in (2.2), we have

*x*

_{2}, · · ·,

*x*

_{ n }∈

*X*. Setting

*y*

_{1}=

*y*

_{2}= 0 and

*x*

_{3}=

*y*

_{3}= · · · =

*x*

_{ n }=

*y*

_{ n }= 0 in (2.2), we have

for all *x*_{1}, *x*_{2} ∈ *X*. Similarly, we get *f*(*x*_{1}, 0, *x*_{3}, 0, · · ·, 0) = · · · = *f*(0, · · ·, 0, *x*_{n-1}, *x*_{
n
} ) = 0 for all *x*_{1}, · · ·, *x*_{
n
} ∈ *X*.

Continuing this process, we obtain that *f*(*x*_{1}, · · ·, *x*_{
n
} ) = 0 for all *x*_{1}, · · ·, *x*_{
n
} ∈ *X* with *x*_{
i
} = 0 for some *i* = 1, · · ·, *n*. Letting *y*_{2} = · · · = *y*_{
n
} = 0 in (2.2), we get the additivity in the first variable. Similarly, the additivities in the remaining variables hold.

The converse is obvious. □

We investigate the generalized Ulam's stability in *C**-ternary algebras for the *n*-additive mappings satisfying (2.1).

**Lemma 2.2**.

*Let X and Y be complex vector spaces and let f*:

*X*

^{ n }→

*Y be a n-additive mapping such that*

*for all*
*and all x*_{1}, · · ·, *x*_{
n
} ∈ *X*, *then f is n-linear over* ℂ.

*Proof*. Since

*f*is

*n*-additive, we get for all

*x*

_{1}, · · ·,

*x*

_{ n }∈

*X*. Now let and

*M*be an integer greater than 2(|

*σ*

_{1}| + · · · + |

*σ*

_{ n }|). Since , there is such that . Now

for all *x*_{1}, · · ·, *x*_{
n
} ∈ *X*. Hence, the mapping *f* : *X*^{
n
} → *Y* is *n*-linear over ℂ. □

Using the above lemma, one can obtain the following result.

**Theorem 2.3**.

*Let X and Y be complex vector spaces and let f*:

*X*

^{ n }→

*Y be a mapping such that*

*for all*
*and all x*_{1,1}, *x*_{2,1}, · · ·, *x*_{1,n}, *x*_{2,n}∈ *X*. *Then*, *f is n-linear over* ℂ.

*Proof*. Letting *λ*_{1} = · · · = *λ*_{
n
} = 1, by Theorem 1.1, *f* is *n*-additive. Letting *x*_{2,1} = · · · = *x*_{2,n}= 0 in (2.3), we get *f*(*λ*_{1}*x*_{1}, · · ·, *λ*_{
n
}*x*_{
n
} ) = *λ*_{1} · · · *λ*_{
n
}*f*(*x*_{1}, · · ·, *x*_{
n
} ) for all
and all *x*_{1}, · · ·, *x*_{
n
} ∈ *X*. Hence, by Lemma 2.2, the mapping *f* is *n*-linear over ℂ. □

From now on, assume that *A* is a *C**-ternary algebra *with* norm || · || _{
A
} and that *B* is a *C**-ternary algebra *with* norm || · ||_{
B
}.

for all
and all *x*_{1,1}, *x*_{2,1}, · · ·, *x*_{1,n}, *x*_{2,n}∈ *A*.

*C**-ternary algebras for the functional equation

**Theorem 2.4**.

*Let p*

_{1}, ···,

*p*

_{ n }∈ (0, ∞)

*with*

*and θ*∈ (0, ∞),

*and let f*:

*A*

^{ n }→

*B be a mapping such that*

*for all*

*and all x*

_{1},

*y*

_{1},

*z*

_{1}, · · ·,

*x*

_{ n },

*y*

_{ n },

*z*

_{ n }∈

*A*.

*Then, there exists a unique C**-

*ternary algebra n-homomorphism H*:

*A*

^{ n }→

*B such that*

*for all x*_{1}, · · ·, *x*_{
n
} ∈ *A*.

*Proof*. Letting

*λ*

_{1}= · · · =

*λ*

_{ n }= 1,

*y*

_{1}=

*x*

_{1}, · · ·,

*y*

_{ n }=

*x*

_{ n }in (2.4), we gain

*x*

_{1}, · · ·,

*x*

_{ n }∈

*A*and all

*j*∈ ℕ. For given integer

*l*,

*m*(0 ≤

*l*<

*m*), we obtain that

*x*

_{1}, ···,

*x*

_{ n }∈

*A*. Since

*B*is complete, the sequence converges for all

*x*

_{1}, · · ·,

*x*

_{ n }∈

*A*. Define

*H*:

*A*

^{ n }→

*B*by

*x*

_{1}, · · ·,

*x*

_{ n }∈

*A*. Letting

*l*= 0 and taking

*m*→ ∞ in (2.8), one can obtain the inequality (2.6). By (2.4), we see that

for all *x*_{1}, *y*_{1}, · · ·, *x*_{
n
} , *y*_{
n
} ∈ *A* and all *s*. Since
, letting *s* → ∞ in the above inequality, *H* satisfies (2.1). By Theorem 2.1, *H* is *n*-additive.

*x*

_{1}, ···,

*x*

_{ n }∈

*A*. From Lemma 2.2, the mapping

*H*:

*A*

^{ n }→

*B*is

*n*-linear over ℂ. It follows from (2.5) that

for all *x*_{1}, *y*_{1}, *z*_{1}, ···, *x*_{
n
} , *y*_{
n
} , *z*_{
n
} ∈ *A*.

which tends to zero as *m* → ∞ for all *x*_{1}, ···, *x*_{
n
} ∈ *A*. Hence, we can conclude that *H*(*x*_{1}, ···, *x*_{
n
} ) = *T*(*x*_{1}, ···, *x*_{
n
} ) for all *x*_{1}, ···, *x*_{
n
} ∈ *A*. This proves the uniqueness of *H*.

Thus, the mapping *H* : *A*→*B* is a unique *C**-ternary algebra *n*-homomorphism satisfying (2.6). □

Letting *p*_{1} = ··· = *p*_{
n
} = 0 and *θ* = *ε* in Theorem 2.4, we obtain the Ulam-Hyers stability for the *n*-additive functional equation (2.1).

*for all*

*and all x*

_{1},

*y*

_{1},

*z*

_{1}···,

*x*

_{ n },

*y*

_{ n },

*z*

_{ n }∈

*A*.

*Then, there exists a unique*C*-

*ternary algebra n-homomorphism*H : A

^{n}→ B

*such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

**Example 2.6**. *We present the following counterexample modiffied by the well-known counterexample of Z. Gajda*[14]*for the functional equation* (2.1). *Set* θ > 0 *and let*
.

*for all x* ∈ ℝ.

*for all*x, y ∈ ℝ,

*that is*, (2.9)

*holds for*m = 1.

*For a ffixed k*∈ ℕ,

*assume that*(2.9)

*holds for*m = k.

*Then, we have*

*for all*
, *that is*, (2.9) *holds for m* = *k* + 1. *Hence*, (2.9) *holds for all m* ∈ ℝ.

*for all* *x*_{1}, ···, *x*_{
n
}∈ ℝ. *Hence, the function* f *is a counterexample for the singular case*
*of Theorem* 2.4.

*for all*

*and all*

*x*

_{1},

*y*

_{1},

*z*

_{1}···,

*x*

_{ n },

*y*

_{ n },

*z*

_{ n }∈

*A*.

*Then, there exists a unique*C*-

*ternary algebra n-homomorphism H*:

*A*

^{ n }→

*B such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

*Proof*. The proof is similar to the proof of Theorem 2.4. □

**Example 2.8**. *We present the following counterexample modiffied by the well-known counterexample of Z. Gajda*[14]*for the functional equation* (2.1). *Set* θ > 0 *and let*

*Let f*: ℝ

^{n}→ ℝ

*and g*: ℝ → ℝ

*be the same as in Example*2.6.

*By the same argument as in Example*2.6,

*for all m*∈ ℕ

*and all*,

*one can obtain that*g

*satisffies the inequality*:

*for all x*

_{1},

*y*

_{1}, ···,

*x*

_{ n },

*y*

_{ n }∈ ℝ.

*For each x*

_{1},

*y*

_{1}, ···,

*x*

_{ n },

*y*

_{ n }∈ ℝ,

*let*M(

*x*

_{1},

*y*

_{1}, ···,

*x*

_{ n },

*y*

_{ n }) := max{|

*x*

_{1}|, |

*y*

_{1}|, ···, |

*x*

_{ n }|, |

*y*

_{ n }|}.

*We have*

*for all* *x*_{1}, *y*_{1}, ···, *x*_{
n
}, *y*_{
n
}∈ ℝ. *By the same reason as for Example* 2.6, *the function f is a counterexample for the singular case p = n of Theorem 2.7*.

**Theorem 2.9**.

*Let p*

_{1}, ···,

*p*

_{ n }∈ (0, ∞)

*with*,

*s*∈ (0,

*n*)

*and θ*,

*η*∈ (0, ∞),

*and let f*:

*A*

^{ n }→

*B be a mapping such that*

*for all*

*and all*

*x*

_{1,1},

*x*

_{2,1},

*x*

_{3,1}, ···,

*x*

_{1, n},

*x*

_{2, n},

*x*

_{3, n}∈

*A*.

*Then, there exists a unique C*-ternary algebra n-homomorphism H: A*

^{ n }→

*B such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

*Proof*. The proof is similar to the proof of Theorem 2.4. □

**Theorem 2.10**.

*Let p*

_{1}, ···,

*p*

_{ n }∈ (0, ∞)

*with*

*and θ*∈ (0, ∞),

*and let f*:

*A*

^{ n }→

*B be a mapping satisfying*(2.4) (2.5).

*Then, there exists a unique C**-

*ternary algebra n-homomorphism H*:

*A*

^{ n }→

*B such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

for all nonnegative integers *m* and *l with m* > *l* and all *x*_{1}, ···, *x*_{
n
} ∈ *A*. It follows from (2.16) that the sequence
is a Cauchy sequence for all *x*_{1}, ···, *x*_{
n
} ∈ *A*. Since *B* is complete, the sequence
converges. Hence, one can define the mapping *H* : *A*^{
n
} → *B* by
for all *x*_{1}, ···, *x*_{
n
} ∈ *A*. Moreover, letting *l* = 0 and passing the limit *m* → ∞ in (2.16), we get (2.15).

The remainder of the proof is similar to the proof of Theorem 2.4. □

**Theorem 2.11**.

*Let p*∈ (3

*n*, ∞)

*and θ*∈ (0, ∞),

*and let f*:

*A*

^{ n }→

*B be a mapping satisfying*(2.11) (2.7),

*and f*(0, ···, 0) = 0.

*Then, there exists a unique C**-

*ternary algebra n-homomorphism H*:

*A*

^{ n }→

*B such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

*Proof*. The proof is similar to that of Theorem 2.10. □

**Theorem 2.12**.

*Let p*

_{1}, ···,

*p*

_{ n }∈ (0, ∞)

*with*,

*s*∈ (

*n*, ∞)

*and θ*,

*η*∈ (0, ∞),

*and let f*:

*A*

^{ n }→

*B be a mapping such that*(2.13), (2.14),

*and f*(0, ···, 0) = 0.

*Then, there exists a unique C**-

*ternary algebra n-homomorphism H*:

*A*

^{ n }→

*B such that*

*for all x*_{1}, ···, *x*_{
n
} ∈ *A*.

*Proof*. The proof is similar to that of Theorem 2.10.

## Declarations

### Acknowledgements

The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.

## Authors’ Affiliations

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